Modeling club structures as bipartite directed networks, we formulate the problem of club formation with multiple memberships as a noncooperative game of network formation and identify conditions on network formation rules and players' network payoffs sufficient to guarantee that the game has a potential function. Our sufficient conditions require that each player choose freely and unilaterally those clubs he joins and also his activities within those clubs (subject to feasibility constraints). We refer to our conditions on rules as noncooperative free mobility. We also require that players' payoffs be additively separable in player-specific payoffs and externalities and that payoff externalities -- a function of club membership, club activities, and crowding -- be identical across players (externality homogeneity). We then show that under these conditions, the noncooperative game of club network formation with multiple memberships is a potential game over directed club networks and we discuss the implications of this result.

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